Question

$ {C_p} - {C_v} $ for an ideal gas is $ R $ . State whether the statement is true or false.

Answer 1

Hint :For a mole of an ideal gas, change in enthalpy can be written in terms of internal energy and work done. Then substituting these values in terms of $ {C_p} $ and $ {C_v} $ , we can deduce the expression for $ {C_p} - {C_v} $ .
 $ H = U + pV $

Complete Step By Step Answer:
At constant pressure and constant volume, the heat capacities are denoted by $ {C_p} $ and $ {C_v} $ respectively. Therefore, the heat at constant pressure can be defined as:
  $ {q_p} = {C_p} \cdot \Delta T $ ; where $ T $ is absolute temperature.
We know that in a chemical change, the released heat at constant pressure is defined as change in enthalpy. Thus,
 $ \Delta H = {q_p} = {C_p} \cdot \Delta T $
The heat at constant pressure can be defined as:
  $ {q_v} = {C_v} \cdot \Delta T $
We know that in a chemical change, the released heat at constant volume is defined as change in internal energy. Thus,
 $ \Delta U = {q_v} = {C_v} \cdot \Delta T $
Now, for a mole of ideal gas, enthalpy is equal to the sum of internal energy and the product of pressure and volume:
 $ H = U + pV $ ; where $ p $ is pressure and $ V $ is volume.
The expression for change in these quantities will be:
 $ \Delta H = \Delta U + \Delta (pV) $
From ideal gas equation, substituting $ pV = RT $ in this expression:
 $ \Delta H = \Delta U + \Delta (RT) $
 $ R $ is universal gas constant, thus:
 $ \Delta H = \Delta U + R\Delta T $
Substituting the values of $ \Delta H $ and $ \Delta U $ in the above expression, we get:
 $ {C_p}\Delta T = {C_v}\Delta T + R\Delta T $
Dividing this equation by $ \Delta T $ , we get:
 $ {C_p} = {C_v} + R $
Rearranging the above expression:
 $ {C_p} - {C_v} = R $
Hence, the given statement is true.

Note :
We should remember the definitions of enthalpy and internal energy. Enthalpy is defined at constant pressure and internal energy is defined at constant volume. Also, the given expression is true only for an ideal gas because we have used the expression for one mole of ideal gas, $ pV = RT $ .